Certifying multipartite entangled states is a basic task in quantum information processing, but the achievable copy complexity depends crucially on the measurements available to the verifier. The strongest possible certification measurement for a known pure target state $|ψ\rangle$ is the two-outcome projector $\{|ψ\rangle\langleψ|,\mathsf{I}-|ψ\rangle\langleψ|\}$ , which is copy-optimal but often experimentally unrealistic or outside the intended measurement model. In this work, we introduce BM-Cert, a single-copy verification protocol for the $n$ -qubit Greenberger--Horne--Zeilinger (GHZ) state using only disjoint two-qubit Bell-basis measurements, together with one single-qubit $X$ -basis measurement when $n$ is odd. Surprisingly, a simple combinatorial effect yields perfect completeness and a verification spectral gap $ν_\mathrm{BM}(n)=1-O(1/n)$ , so our depth-2 protocol already approaches the ideal projective verification asymptotically as $n$ grows. This contrasts with local Pauli GHZ verification, whose optimal spectral gap remains bounded away from $1$ . Thus, allowing only two-qubit entangling measurements on disjoint pairs is enough to achieve asymptotically ideal projective certification. The same Bell-matching outcomes also yield BM-Fid, an unbiased estimator of the GHZ fidelity whose leading Hoeffding coefficient in the sample complexity tends to the ideal value achieved by direct projection. For the open-boundary linear nearest neighbor setting, we further introduce Brick-Cert, a disjoint 2-local certification protocol whose spectral gap $4/5$ is optimal within that restricted architecture.Certifying multipartite entangled states is a basic task in quantum information processing, but the achievable copy complexity depends crucially on the measurements available to the verifier. The strongest possible certification measurement for a known pure target state $|ψ\rangle$ is the two-outcome projector $\{|ψ\rangle\langleψ|,\mathsf{I}-|ψ\rangle\langleψ|\}$, which is copy-optimal but often experimentally unrealistic or outside the intended measurement model. In this work, we introduce BM-Cert, a single-copy verification protocol for the $n$-qubit Greenberger--Horne--Zeilinger (GHZ) state using only disjoint two-qubit Bell-basis measurements, together with one single-qubit $X$-basis measurement when $n$ is odd. Surprisingly, a simple combinatorial effect yields perfect completeness and a verification spectral gap $ν_\mathrm{BM}(n)=1-O(1/n)$, so our depth-2 protocol already approaches the ideal projective verification asymptotically as $n$ grows. This contrasts with local Pauli GHZ verification, whose optimal spectral gap remains bounded away from $1$. Thus, allowing only two-qubit entangling measurements on disjoint pairs is enough to achieve asymptotically ideal projective certification. The same Bell-matching outcomes also yield BM-Fid, an unbiased estimator of the GHZ fidelity whose leading Hoeffding coefficient in the sample complexity tends to the ideal value achieved by direct projection. For the open-boundary linear nearest neighbor setting, we further introduce Brick-Cert, a disjoint 2-local certification protocol whose spectral gap $4/5$ is optimal within that restricted architecture.

quant-pharXiv:2606.09947Jun 9, 20260 citations

Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion*

L Brightmore, G P Gehér, Alexander Rudol'fovich Its +4 more

Abstract We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv: 1104.1004 , for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m} ∪ {2m + 1, 2m + 2, …, 3m}, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.Abstract We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv: 1104.1004 , for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m} ∪ {2m + 1, 2m + 2, …, 3m}, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.

arXiv:1912.08658Journal of Physics A Mathematical and TheoreticalJun 15, 20208 citations

Bounded Quantum Systems

L. E. Reichl

Institute for Nonlinear ScienceJan 1, 20040 citations

Entangled Quantum States

Willi-Hans Steeb, Yorick Hardy

International Journal of Theoretical PhysicsDec 1, 20002 citations

Modular Nuclearity and Entanglement measures

Lorenzo Panebianco, Benedikt Wegener

In the framework of Algebraic Quantum Field Theory, several operator algebraic notions of entanglement entropy can be associated to a couple of causally disjoint and distant spacetime regions $\mathcal{S}_A$ and $\mathcal{S}_B$ . In this work, we show that the mutual information is finite in any local QFT verifying a modular $p$ -nuclearity condition for some $0 < p <1$ . A similar result is proved for another recently studied entanglement measure. Furthermore, if we assume conformal covariance then by comparison with other entanglement measures we can state that the mutual information satisfies lower bounds of area law type when the distance between $\mathcal{S}_A$ and $\mathcal{S}_B$ approaches to zero. As an application, in $1+1$ -dimensional integrable models with factorizing S-matrices, we study the asymptotic behaviour of different entanglement measures as the distance between two causally disjoint wedges diverges.In the framework of Algebraic Quantum Field Theory, several operator algebraic notions of entanglement entropy can be associated to a couple of causally disjoint and distant spacetime regions $\mathcal{S}_A$ and $\mathcal{S}_B$. In this work, we show that the mutual information is finite in any local QFT verifying a modular $p$-nuclearity condition for some $0 < p <1$. A similar result is proved for another recently studied entanglement measure. Furthermore, if we assume conformal covariance then by comparison with other entanglement measures we can state that the mutual information satisfies lower bounds of area law type when the distance between $\mathcal{S}_A$ and $\mathcal{S}_B$ approaches to zero. As an application, in $1+1$-dimensional integrable models with factorizing S-matrices, we study the asymptotic behaviour of different entanglement measures as the distance between two causally disjoint wedges diverges.

arXiv:2110.05823Oct 12, 20210 citations

|ψ\rangle

is the two-outcome projector

\{|ψ\rangle\langleψ|,\mathsf{I}-|ψ\rangle\langleψ|\}

, which is copy-optimal but often experimentally unrealistic or outside the intended measurement model. In this work, we introduce BM-Cert, a single-copy verification protocol for the

n

-qubit Greenberger--Horne--Zeilinger (GHZ) state using only disjoint two-qubit Bell-basis measurements, together with one single-qubit

X

-basis measurement when

n

is odd. Surprisingly, a simple combinatorial effect yields perfect completeness and a verification spectral gap

ν_\mathrm{BM}(n)=1-O(1/n)

, so our depth-2 protocol already approaches the ideal projective verification asymptotically as

n

grows. This contrasts with local Pauli GHZ verification, whose optimal spectral gap remains bounded away from

1

. Thus, allowing only two-qubit entangling measurements on disjoint pairs is enough to achieve asymptotically ideal projective certification. The same Bell-matching outcomes also yield BM-Fid, an unbiased estimator of the GHZ fidelity whose leading Hoeffding coefficient in the sample complexity tends to the ideal value achieved by direct projection. For the open-boundary linear nearest neighbor setting, we further introduce Brick-Cert, a disjoint 2-local certification protocol whose spectral gap

4/5

is optimal within that restricted architecture.

In the framework of Algebraic Quantum Field Theory, several operator algebraic notions of entanglement entropy can be associated to a couple of causally disjoint and distant spacetime regions

\mathcal{S}_A

and

\mathcal{S}_B

. In this work, we show that the mutual information is finite in any local QFT verifying a modular

p

-nuclearity condition for some

0 < p <1

. A similar result is proved for another recently studied entanglement measure. Furthermore, if we assume conformal covariance then by comparison with other entanglement measures we can state that the mutual information satisfies lower bounds of area law type when the distance between

\mathcal{S}_A

and

\mathcal{S}_B

approaches to zero. As an application, in

1+1

-dimensional integrable models with factorizing S-matrices, we study the asymptotic behaviour of different entanglement measures as the distance between two causally disjoint wedges diverges.